Which of the following is true for f(x) (a) f(x) is one-one...

Which of the following is true for $f(x)$
(a) $f(x)$ is one-one and into
(b) $\{f(x)\}$ is non-periodic, where \{\} denotes fractional part of $x$.
(c) $f(x)=4$ has only two solutions.
(d) $f(x)=f^{-1}(x)$ has only one solution.

Key Concepts

Solving Functional Equations

Solving equations involving functions, such as f(x) = constant or f(x) = f?¹(x), is a critical concept that combines the understanding of a function's properties with algebraic techniques. This includes determining the number of solutions and analyzing how the function’s inherent characteristics, like injectivity or periodicity, influence the existence and uniqueness of these solutions.

Fractional Part Function and Periodicity

The fractional part function, typically denoted by braces as {x}, extracts the non?integer part of a real number. A key characteristic of this function is its periodicity, specifically with a period of 1, because the fractional parts of x and x+1 are identical. Analyzing the periodic behavior of functions involving the fractional part is an important concept when considering the overall periodic nature of composite or modified functions.

Inverse Functions and Their Relationship to the Original Function

An inverse function reverses the effect of the original function by mapping outputs back to inputs. For a function to have an inverse, it must be both injective and, oftentimes, surjective. The equation f(x) = f?¹(x) explores the relationship between a function and its inverse, frequently revealing symmetries—especially symmetry with respect to the line y = x—and fixed points where a function equals its own inverse.

Injective (One?one) Functions

An injective function, or one?one function, is defined as a mapping where distinct inputs always yield distinct outputs. In other words, if f(a) equals f(b), then it must be true that a equals b. This property is fundamental in function theory and is essential for determining the possibility of defining an inverse function, since only injective functions have unique pre?images for every element in their image.

Surjective (Onto) Functions

A surjective function, often called an 'onto' function, ensures that every element in the function’s codomain is the image of at least one element from its domain. This concept is important when analyzing the range of the function relative to the codomain, and it plays a crucial role in understanding how completely the function covers the target set.

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